The matching extendability of optimal $1$-embedded graphs on the projective plane

TL;DR

The paper addresses the problem of matching extendability for optimal $1$-embedded graphs on the projective plane (O1PPG). It leverages the quadrangulation $Q(G)$ of non-crossing edges, barrier cycle analysis, and minimal-cut structure, together with a key topological input that every $4$-connected graph on $P^2$ is Hamilton-connected. The results show that all even-order O1PPG are $1$-extendable, characterize $2$-extendability in terms of barrier $4$-cycles, and precisely describe obstructions to $3$-extendability for $5$-connected O1PPG (including an odd weighted region bound by a $6$-cycle and a family of $Q(G)$-subgraphs). The work also identifies a projective-bowtie structure governing $6$-connectivity and concludes with remarks and conjectures about extendability on other closed surfaces, highlighting both similarities and key differences from the spherical case.

Abstract

In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n- 4$ edges. We first show that every O1PPG of even order is $1$-extendable. Next, we characterize $2$-extendable O1PPG's in terms of a separating cycle consisting of only non-crossing edges. Moreover, we characterize O1PPG's having connectivity exactly $5$. Using the characterization, we further identify three independent edges in those graphs that are not extendable.

Figures (5)

• Figure 1: Specified subgraphs in $Q(G)$ in Theorem \ref{['thm:3-ext']}.
• Figure 2: Projective-bowtie
• Figure 3: $Q[S]$ obtained by a minimal $6$-cut $S$.
• Figure 4: $Q[S]$ in the case of $|S| = 7$.
• Figure 5: Adding $v$ of degree $2$ inside the $8$-gonal region.